Integrand size = 24, antiderivative size = 250 \[ \int (a+b x)^{3/2} (A+B x) \sqrt {d+e x} \, dx=\frac {(b d-a e)^2 (5 b B d-8 A b e+3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^2 e^3}-\frac {(b d-a e) (5 b B d-8 A b e+3 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{96 b^2 e^2}-\frac {(5 b B d-8 A b e+3 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{24 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{3/2}}{4 b e}-\frac {(b d-a e)^3 (5 b B d-8 A b e+3 a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{5/2} e^{7/2}} \]
1/4*B*(b*x+a)^(5/2)*(e*x+d)^(3/2)/b/e-1/64*(-a*e+b*d)^3*(-8*A*b*e+3*B*a*e+ 5*B*b*d)*arctanh(e^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(e*x+d)^(1/2))/b^(5/2)/e^(7 /2)-1/96*(-a*e+b*d)*(-8*A*b*e+3*B*a*e+5*B*b*d)*(b*x+a)^(3/2)*(e*x+d)^(1/2) /b^2/e^2-1/24*(-8*A*b*e+3*B*a*e+5*B*b*d)*(b*x+a)^(5/2)*(e*x+d)^(1/2)/b^2/e +1/64*(-a*e+b*d)^2*(-8*A*b*e+3*B*a*e+5*B*b*d)*(b*x+a)^(1/2)*(e*x+d)^(1/2)/ b^2/e^3
Time = 0.66 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.93 \[ \int (a+b x)^{3/2} (A+B x) \sqrt {d+e x} \, dx=\frac {\sqrt {a+b x} \sqrt {d+e x} \left (-9 a^3 B e^3+3 a^2 b e^2 (3 B d+8 A e+2 B e x)+a b^2 e \left (16 A e (4 d+7 e x)+B \left (-31 d^2+20 d e x+72 e^2 x^2\right )\right )+b^3 \left (8 A e \left (-3 d^2+2 d e x+8 e^2 x^2\right )+B \left (15 d^3-10 d^2 e x+8 d e^2 x^2+48 e^3 x^3\right )\right )\right )}{192 b^2 e^3}+\frac {(b d-a e)^3 (-5 b B d+8 A b e-3 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{64 b^{5/2} e^{7/2}} \]
(Sqrt[a + b*x]*Sqrt[d + e*x]*(-9*a^3*B*e^3 + 3*a^2*b*e^2*(3*B*d + 8*A*e + 2*B*e*x) + a*b^2*e*(16*A*e*(4*d + 7*e*x) + B*(-31*d^2 + 20*d*e*x + 72*e^2* x^2)) + b^3*(8*A*e*(-3*d^2 + 2*d*e*x + 8*e^2*x^2) + B*(15*d^3 - 10*d^2*e*x + 8*d*e^2*x^2 + 48*e^3*x^3))))/(192*b^2*e^3) + ((b*d - a*e)^3*(-5*b*B*d + 8*A*b*e - 3*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a + b*x] )])/(64*b^(5/2)*e^(7/2))
Time = 0.26 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.84, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {90, 60, 60, 60, 66, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (a+b x)^{3/2} (A+B x) \sqrt {d+e x} \, dx\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {B (a+b x)^{5/2} (d+e x)^{3/2}}{4 b e}-\frac {(3 a B e-8 A b e+5 b B d) \int (a+b x)^{3/2} \sqrt {d+e x}dx}{8 b e}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {B (a+b x)^{5/2} (d+e x)^{3/2}}{4 b e}-\frac {(3 a B e-8 A b e+5 b B d) \left (\frac {(b d-a e) \int \frac {(a+b x)^{3/2}}{\sqrt {d+e x}}dx}{6 b}+\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 b}\right )}{8 b e}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {B (a+b x)^{5/2} (d+e x)^{3/2}}{4 b e}-\frac {(3 a B e-8 A b e+5 b B d) \left (\frac {(b d-a e) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}}dx}{4 e}\right )}{6 b}+\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 b}\right )}{8 b e}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {B (a+b x)^{5/2} (d+e x)^{3/2}}{4 b e}-\frac {(3 a B e-8 A b e+5 b B d) \left (\frac {(b d-a e) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}}dx}{2 e}\right )}{4 e}\right )}{6 b}+\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 b}\right )}{8 b e}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {B (a+b x)^{5/2} (d+e x)^{3/2}}{4 b e}-\frac {(3 a B e-8 A b e+5 b B d) \left (\frac {(b d-a e) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \int \frac {1}{b-\frac {e (a+b x)}{d+e x}}d\frac {\sqrt {a+b x}}{\sqrt {d+e x}}}{e}\right )}{4 e}\right )}{6 b}+\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 b}\right )}{8 b e}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {B (a+b x)^{5/2} (d+e x)^{3/2}}{4 b e}-\frac {(3 a B e-8 A b e+5 b B d) \left (\frac {(b d-a e) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} e^{3/2}}\right )}{4 e}\right )}{6 b}+\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 b}\right )}{8 b e}\) |
(B*(a + b*x)^(5/2)*(d + e*x)^(3/2))/(4*b*e) - ((5*b*B*d - 8*A*b*e + 3*a*B* e)*(((a + b*x)^(5/2)*Sqrt[d + e*x])/(3*b) + ((b*d - a*e)*(((a + b*x)^(3/2) *Sqrt[d + e*x])/(2*e) - (3*(b*d - a*e)*((Sqrt[a + b*x]*Sqrt[d + e*x])/e - ((b*d - a*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(Sq rt[b]*e^(3/2))))/(4*e)))/(6*b)))/(8*b*e)
3.23.13.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Leaf count of result is larger than twice the leaf count of optimal. \(967\) vs. \(2(212)=424\).
Time = 1.09 (sec) , antiderivative size = 968, normalized size of antiderivative = 3.87
| method | result | size |
| default | \(-\frac {\sqrt {b x +a}\, \sqrt {e x +d}\, \left (-40 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a \,b^{2} d \,e^{2} x -9 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{4} e^{4}+15 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{4} d^{4}+48 A \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b^{3} d^{2} e -48 A \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a^{2} b \,e^{3}-36 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{3} d^{3} e +18 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} b^{2} d^{2} e^{2}-72 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} b^{2} d \,e^{3}+72 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{3} d^{2} e^{2}+12 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{3} b d \,e^{3}-18 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a^{2} b d \,e^{2}+62 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a \,b^{2} d^{2} e +18 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a^{3} e^{3}-30 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b^{3} d^{3}+24 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{3} b \,e^{4}-24 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{4} d^{3} e -224 A \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a \,b^{2} e^{3} x -32 A \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b^{3} d \,e^{2} x -12 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a^{2} b \,e^{3} x +20 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b^{3} d^{2} e x -144 B a \,b^{2} e^{3} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-16 B \,b^{3} d \,e^{2} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-128 A a \,b^{2} d \,e^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-96 B \,b^{3} e^{3} x^{3} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-128 A \,b^{3} e^{3} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\right )}{384 b^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, e^{3} \sqrt {b e}}\) | \(968\) |
-1/384*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(-40*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1 /2)*a*b^2*d*e^2*x-9*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2 )+a*e+b*d)/(b*e)^(1/2))*a^4*e^4+15*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^( 1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^4*d^4+48*A*((b*x+a)*(e*x+d))^(1/2 )*(b*e)^(1/2)*b^3*d^2*e-48*A*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a^2*b*e^3 -36*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e) ^(1/2))*a*b^3*d^3*e+18*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^( 1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b^2*d^2*e^2-72*A*ln(1/2*(2*b*e*x+2*((b*x+a) *(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b^2*d*e^3+72*A*ln(1/ 2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b ^3*d^2*e^2+12*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+ b*d)/(b*e)^(1/2))*a^3*b*d*e^3-18*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a^2 *b*d*e^2+62*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a*b^2*d^2*e+18*B*((b*x+a )*(e*x+d))^(1/2)*(b*e)^(1/2)*a^3*e^3-30*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1 /2)*b^3*d^3+24*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e +b*d)/(b*e)^(1/2))*a^3*b*e^4-24*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2 )*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^4*d^3*e-224*A*((b*x+a)*(e*x+d))^(1/2 )*(b*e)^(1/2)*a*b^2*e^3*x-32*A*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*b^3*d*e ^2*x-12*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a^2*b*e^3*x+20*B*((b*x+a)*(e *x+d))^(1/2)*(b*e)^(1/2)*b^3*d^2*e*x-144*B*a*b^2*e^3*x^2*((b*x+a)*(e*x+...
Time = 0.26 (sec) , antiderivative size = 766, normalized size of antiderivative = 3.06 \[ \int (a+b x)^{3/2} (A+B x) \sqrt {d+e x} \, dx=\left [\frac {3 \, {\left (5 \, B b^{4} d^{4} - 4 \, {\left (3 \, B a b^{3} + 2 \, A b^{4}\right )} d^{3} e + 6 \, {\left (B a^{2} b^{2} + 4 \, A a b^{3}\right )} d^{2} e^{2} + 4 \, {\left (B a^{3} b - 6 \, A a^{2} b^{2}\right )} d e^{3} - {\left (3 \, B a^{4} - 8 \, A a^{3} b\right )} e^{4}\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} - 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) + 4 \, {\left (48 \, B b^{4} e^{4} x^{3} + 15 \, B b^{4} d^{3} e - {\left (31 \, B a b^{3} + 24 \, A b^{4}\right )} d^{2} e^{2} + {\left (9 \, B a^{2} b^{2} + 64 \, A a b^{3}\right )} d e^{3} - 3 \, {\left (3 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} e^{4} + 8 \, {\left (B b^{4} d e^{3} + {\left (9 \, B a b^{3} + 8 \, A b^{4}\right )} e^{4}\right )} x^{2} - 2 \, {\left (5 \, B b^{4} d^{2} e^{2} - 2 \, {\left (5 \, B a b^{3} + 4 \, A b^{4}\right )} d e^{3} - {\left (3 \, B a^{2} b^{2} + 56 \, A a b^{3}\right )} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{768 \, b^{3} e^{4}}, \frac {3 \, {\left (5 \, B b^{4} d^{4} - 4 \, {\left (3 \, B a b^{3} + 2 \, A b^{4}\right )} d^{3} e + 6 \, {\left (B a^{2} b^{2} + 4 \, A a b^{3}\right )} d^{2} e^{2} + 4 \, {\left (B a^{3} b - 6 \, A a^{2} b^{2}\right )} d e^{3} - {\left (3 \, B a^{4} - 8 \, A a^{3} b\right )} e^{4}\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) + 2 \, {\left (48 \, B b^{4} e^{4} x^{3} + 15 \, B b^{4} d^{3} e - {\left (31 \, B a b^{3} + 24 \, A b^{4}\right )} d^{2} e^{2} + {\left (9 \, B a^{2} b^{2} + 64 \, A a b^{3}\right )} d e^{3} - 3 \, {\left (3 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} e^{4} + 8 \, {\left (B b^{4} d e^{3} + {\left (9 \, B a b^{3} + 8 \, A b^{4}\right )} e^{4}\right )} x^{2} - 2 \, {\left (5 \, B b^{4} d^{2} e^{2} - 2 \, {\left (5 \, B a b^{3} + 4 \, A b^{4}\right )} d e^{3} - {\left (3 \, B a^{2} b^{2} + 56 \, A a b^{3}\right )} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{384 \, b^{3} e^{4}}\right ] \]
[1/768*(3*(5*B*b^4*d^4 - 4*(3*B*a*b^3 + 2*A*b^4)*d^3*e + 6*(B*a^2*b^2 + 4* A*a*b^3)*d^2*e^2 + 4*(B*a^3*b - 6*A*a^2*b^2)*d*e^3 - (3*B*a^4 - 8*A*a^3*b) *e^4)*sqrt(b*e)*log(8*b^2*e^2*x^2 + b^2*d^2 + 6*a*b*d*e + a^2*e^2 - 4*(2*b *e*x + b*d + a*e)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 8*(b^2*d*e + a*b *e^2)*x) + 4*(48*B*b^4*e^4*x^3 + 15*B*b^4*d^3*e - (31*B*a*b^3 + 24*A*b^4)* d^2*e^2 + (9*B*a^2*b^2 + 64*A*a*b^3)*d*e^3 - 3*(3*B*a^3*b - 8*A*a^2*b^2)*e ^4 + 8*(B*b^4*d*e^3 + (9*B*a*b^3 + 8*A*b^4)*e^4)*x^2 - 2*(5*B*b^4*d^2*e^2 - 2*(5*B*a*b^3 + 4*A*b^4)*d*e^3 - (3*B*a^2*b^2 + 56*A*a*b^3)*e^4)*x)*sqrt( b*x + a)*sqrt(e*x + d))/(b^3*e^4), 1/384*(3*(5*B*b^4*d^4 - 4*(3*B*a*b^3 + 2*A*b^4)*d^3*e + 6*(B*a^2*b^2 + 4*A*a*b^3)*d^2*e^2 + 4*(B*a^3*b - 6*A*a^2* b^2)*d*e^3 - (3*B*a^4 - 8*A*a^3*b)*e^4)*sqrt(-b*e)*arctan(1/2*(2*b*e*x + b *d + a*e)*sqrt(-b*e)*sqrt(b*x + a)*sqrt(e*x + d)/(b^2*e^2*x^2 + a*b*d*e + (b^2*d*e + a*b*e^2)*x)) + 2*(48*B*b^4*e^4*x^3 + 15*B*b^4*d^3*e - (31*B*a*b ^3 + 24*A*b^4)*d^2*e^2 + (9*B*a^2*b^2 + 64*A*a*b^3)*d*e^3 - 3*(3*B*a^3*b - 8*A*a^2*b^2)*e^4 + 8*(B*b^4*d*e^3 + (9*B*a*b^3 + 8*A*b^4)*e^4)*x^2 - 2*(5 *B*b^4*d^2*e^2 - 2*(5*B*a*b^3 + 4*A*b^4)*d*e^3 - (3*B*a^2*b^2 + 56*A*a*b^3 )*e^4)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^3*e^4)]
\[ \int (a+b x)^{3/2} (A+B x) \sqrt {d+e x} \, dx=\int \left (A + B x\right ) \left (a + b x\right )^{\frac {3}{2}} \sqrt {d + e x}\, dx \]
Exception generated. \[ \int (a+b x)^{3/2} (A+B x) \sqrt {d+e x} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Leaf count of result is larger than twice the leaf count of optimal. 1071 vs. \(2 (212) = 424\).
Time = 0.44 (sec) , antiderivative size = 1071, normalized size of antiderivative = 4.28 \[ \int (a+b x)^{3/2} (A+B x) \sqrt {d+e x} \, dx=\text {Too large to display} \]
1/192*(8*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*( 4*(b*x + a)/b^2 + (b^6*d*e^3 - 13*a*b^5*e^4)/(b^7*e^4)) - 3*(b^7*d^2*e^2 + 2*a*b^6*d*e^3 - 11*a^2*b^5*e^4)/(b^7*e^4)) - 3*(b^3*d^3 + a*b^2*d^2*e + 3 *a^2*b*d*e^2 - 5*a^3*e^3)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/(sqrt(b*e)*b*e^2))*A*abs(b) + (sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^3 + (b^12*d*e^5 - 25*a*b^11*e^6)/(b^14*e^6)) - (5*b^13*d^2*e^4 + 14*a*b^12*d*e^5 - 163*a^ 2*b^11*e^6)/(b^14*e^6)) + 3*(5*b^14*d^3*e^3 + 9*a*b^13*d^2*e^4 + 15*a^2*b^ 12*d*e^5 - 93*a^3*b^11*e^6)/(b^14*e^6))*sqrt(b*x + a) + 3*(5*b^4*d^4 + 4*a *b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 - 35*a^4*e^4)*log(abs(-sqr t(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/(sqrt(b*e)*b^ 2*e^3))*B*abs(b) - 192*((b^2*d - a*b*e)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/sqrt(b*e) - sqrt(b^2*d + (b*x + a)* b*e - a*b*e)*sqrt(b*x + a))*A*a^2*abs(b)/b^2 + 16*(sqrt(b^2*d + (b*x + a)* b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*d*e^3 - 13 *a*b^5*e^4)/(b^7*e^4)) - 3*(b^7*d^2*e^2 + 2*a*b^6*d*e^3 - 11*a^2*b^5*e^4)/ (b^7*e^4)) - 3*(b^3*d^3 + a*b^2*d^2*e + 3*a^2*b*d*e^2 - 5*a^3*e^3)*log(abs (-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/(sqrt(b* e)*b*e^2))*B*a*abs(b)/b + 48*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*b*x + 2*a + (b*d*e - 5*a*e^2)/e^2)*sqrt(b*x + a) + (b^3*d^2 + 2*a*b^2*d*e - ...
Timed out. \[ \int (a+b x)^{3/2} (A+B x) \sqrt {d+e x} \, dx=\int \left (A+B\,x\right )\,{\left (a+b\,x\right )}^{3/2}\,\sqrt {d+e\,x} \,d x \]